Cantor diagonal argument's array seems to be with only numbers $\in [0,1]$, but Rudin (Principles of Mathematical Analysis, $2.14$), if I understood well, uses the argument with an array of numbers $\in \mathbb R$.
Here's my problem:
The binary representation of a real number is as such: a finite binary number then, a decimal separator$^1$ then, an infinite binary number.
But, provided the array of Rudin is made of an infinite number of binary representations of real numbers, we don't exactly know where to place the decimal separator in the array.
There are two options: either we place in a common position the decimal separator (first case), either we place the decimal separator at a different position depending on the real number considered (second case).
Now, when taking the "new number" (the one we obtain by switching the $0$'s to $1$'s and the $1$'s to $0$'s in the diagonal process), I can clearly see where the decimal separator of the "new number" will be if we place ourselves in the first case. If we place ourselves in the second case, I have a hard time to see where the decimal separator should be placed in the "new number".
Next, a quite vague question, but I still have it: if the first case is the right case, how do we find the common position of the decimal separator ?
An answer (my answer) to this question would be that the decimal separator is located "very far" on the right of the array. But I am not sure it is a right answer.
Thanks in advance.
$^1$: Perhaps shall I rather use the word binary separator.